A Look at the Inner Structure of the 2-adic Ring $C^*$‡-algebra and Its Automorphism Groups

Abstract

We undertake a systematic study of the so-called 2-adic ring $C^$-algebra $\mathcal Q\_2$. This is the universal $C^$-algebra generated by a unitary $U$ and an isometry $S\_2$ such that $S\_2U=U^2S\_2$ and $S\_2S\_2^+US\_2S\_2^U^=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal O\_2=C^(S\_1, S\_2)$ through the injective homomorphism mapping $S\_1$ to $US\_2$. Among the main results, the relative commutant $C^(S\_2)'\cap \mathcal Q\_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal O\_2\subset\mathcal Q\_2$, namely the endomorphisms of $\mathcal Q\_2$ that restrict to the identity on $\mathcal O\_2$ are actually the identity on the whole $\mathcal Q\_2$. Moreover, there is no conditional expectation from $\mathcal Q\_2$ onto $\mathcal O\_2$. As for the inner structure of $\mathcal Q\_2$, the diagonal subalgebra $\mathcal D\_2$ and $C^(U)$ are both proved to be maximal abelian in $\mathcal Q\_2$. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of $\mathcal Q\_2$. In particular, the semigroup of the endomorphisms fixing $U$ turns out to be a maximal abelian subgroup of Aut$(\mathcal Q\_2)$ topologically isomorphic with $C(\mathbb{T},\mathbb{T})$. Finally, it is shown by an explicit construction that Out$(\mathcal Q\_2)$ is uncountable and non-abelian.